Thermal ripples in bilayer graphene

Abstract

We study thermal fluctuations of freestanding bilayer graphene subject to vanishing external tension. Within a phenomenological theory, the system is described as a stack of two continuum crystalline membranes, characterized by finite elastic moduli and a nonzero bending rigidity. A nonlinear rotationally invariant model guided by elasticity theory is developed to describe interlayer interactions. After neglection of in-plane phonon nonlinearities and anharmonic interactions involving interlayer shear and compression modes, an effective theory for soft flexural fluctuations of the bilayer is constructed. The resulting model, neglecting anisotropic interactions, has the same form of a well-known effective theory for out-of-plane fluctuations in a single-layer membrane, but with a strongly wave-vector-dependent bare bending rigidity. Focusing on AB-stacked bilayer graphene, parameters governing interlayer interactions in the theory are derived by first-principles calculations. Statistical-mechanical properties of interacting flexural fluctuations are then calculated by a numerical iterative solution of field-theory integral equations within the self-consistent screening approximation. The bare bending rigidity in the considered model exhibits a crossover between a long-wavelength regime governed by in-plane elastic stress and a short wavelength region controlled by monolayer curvature stiffness. Interactions between flexural fluctuations drive a further crossover between a harmonic and a strong-coupling regime, characterized by anomalous scale invariance. The overlap and interplay between these two crossover behaviors is analyzed at varying temperatures.

Revealing 3D Ripple Structure and Its Dynamics in Freestanding Monolayer MoSe2 by Single-Frame 2D Atomic Image Reconstruction

“…Phenomenological theories of the thermodynamic behavior of flexible membranes predict a power law for the correlation function of the out-of-plane displacements, namely, ⟨| h q | 2 ⟩ ∝ q – n , where h q is the Fourier component of the rippling height at a wave vector with magnitude q and the bracket ⟨···⟩ denotes an ensemble average. The exponent n is equal to 4 within the harmonic approximation, but it becomes less than 4 and is no longer a constant when considering the coupling between bending and stretching modes. − Monte Carlo and molecular dynamics (MD) simulations have validated the power law in two-dimensional crystalline materials including graphene, h-BN, and MoS 2 . − …”